By Sebastian Hayes

Subitizing : Straight Quantitative Assessments

April 16, 2012

The natives on my imaginary island are innocent of numbers, possessing at most a handful of number words. So, how on earth did they manage to get on all this time without them? Is my claim that you don’t need numbers in a non-commercial society an adequate explanation?
The answer is that, to judge by first-hand accounts of similar peoples, my hypothetical natives undoubtedly had an awareness of quantity which was in certain respects far more refined than ours. But this awareness of comparative size and quantity was not numerical in the accepted sense of the word because it was not linked to verbal or recorded symbols.1 Man is the symbol-making animal par excellence and symbol-making — or symbol-mongering — does seem to be one of the main features which distinguishes the human species from other mammals. Animals do not of their own accord seem to use symbols and the fact that apes can be taught to use them by humans is really neither here nor there1. Also, practically all societies at the hunting/food-gathering stage did not develop numerical symbols to any great extent though they did develop highly complex languages, myths, rituals and the like and in many cases produced superb paintings and sculptures2. But this does not mean that had no awareness of quantity nor any way of expressing differences between different quantities, indeed they would not have survived if they did not have such a sense.
It is well known that a teacher entering a classroom often ‘instinctively’ notices if a single pupil is missing. In this case the teacher obviously does ‘know about numbers’ but he or she does not count the number of pupils present, an extremely laborious task, and would certainly be unable to assess correctly the number of pupils present in a classroom just by glancing at them2. The point is that, if we are familiar with a particular quantity, we notice ‘immediately’ very small deviations from it — one item missing, one or two extra items. This sense is extremely primitive and at the hunting stage was much more developed than it is now3. Holmberg, in his field study of the Siriona of Bolivia writes, “A man who has a hundred ears of corn hanging on a pole….will note the lack of one ear immediately” (Closs, p. 17). A missionary to the Abipones, another South American Indian tribe, witnessed a mass migration. “The long train …..was surrounded on both sides by countless numbers of dogs. From their saddles the Indians would look around and inspect them. If so much as a single dog was missing from the huge pack, they would keep calling until all were collected again” (Menninger, p.11). Yet these Indian tribes had only a handful of number words — for the Siriona “everything beyond 3 was etubiana, ‘much’, or eata, ‘many’ ” (Closs).
Note that the missionary himself describes the pack as made up of ‘countless numbers of dogs’ . Even if the numerate white man had laboriously counted the pack the previous night at the camp site, this numerical knowledge would have been worse than useless once the train was on the move — far more useful was the innumerate but finely tuned sense-perception of the Indian. Although as far as I know no extensive studies have been done on this, it would seem plausible that this ability for ‘direct quantitative assessment’ interferes with ‘number sense’ and vice versa, i.e. those who are good at the one are not good at the other. Certainly, Gay and Cole note how much better the African Kpelle were at ‘quantitative assessments’ such as evaluating how many cups of rice would fill a certain bowl than educated American Peace Corps volunteers4. This would explain the extreme resistance of primitive tribes to acquiring the white man’s numerical skills : the Spanish found they had to impose European number systems by force (backed up by threats of damnation) and in the priests strictly forbade the Indians to use their own number systems if they had them.
Even today when everything is measured, everything has a number attached to it, we still use non-numerical assessment of quantity referring unknown or little known quantities to some well-known amount which is, at least to us, standard. A farmer assesses the size of a farm he is visiting not in acres but by comparing it to the size of his own and might well assess the size of a meadow in terms of the number of cows it would feed, a form of measurement which allows him to take into account qualitative features such as the lushness of the grass, features which the numerical assessment ignores completely. Here land is assessed in terms of animals since that is what the farmer is above all concerned with. In a converse ‘cross-measurement’ the Abipones assessed the size of a herd of horses “by indicating how much space the horses occupied when standing next to each other” (Menninger). Distance to a certain place was not measured in abstract SI units that one cannot visualize or ‘feel’ with one’s body, but in terms of the journey time it took to reach them : such and such a place was “two days sleep away”. This is a far more practical and succinct evaluation than the modern equivalent which might well be, “It’s 50 kilometres away but you’d probably have to make a detour……”
In the world of a hunter/nomad variable quantities are related to a known standard quantity just as they are today with our SI units : the difference is that the standard quantity is, for us, purely objective, remote from our experience, whereas for the hunter/nomad it is a personally experienced quantity that he carries around with him all his life. Even in the very different world of the agriculturalist standard quantities were still connected up to daily life — an acre was originally supposed to be the amount of arable land a ploughman could plough in a day. However, in the strictly scientific metric system introduced (or rather imposed) by Napoleon every trace of subjectivity is deliberately removed and the basic unit of distance, the metre, is no longer related to the human body (cubit, foot &c.) but is fixed as a fraction of the diameter of the Earth.
It is because it is so important psychologically (and hence practically also) to have a set of standard quantities well fixed in our minds, that there has been such entrenched resistance right across the board to the introduction of the metric system. Prior to the change in the law, people in this country had their own Bureau International de Mesures in their minds and ‘at their fingertips’.
People of the present society, so ultra-educated and urban, and especially mathematicians themselves who have a vested interest in their ignorance of other ways of living, fail to realize how utterly irrelevant and indeed counter-productive purely numerical knowledge often is. There is nothing at all absurd about a native mother not knowing numerically speaking how many children she has got — and since most South American Indian women presumably gave birth to many more than three children this state of affairs may well have existed throughout the entire history of ‘innumerable’ tribes incredible though this may sound. For why should she know ‘how many’ children she has when she knows them all individually? Even if she were more numerate than the Abipone she may still not have known because she never saw fit to ask the question ‘how many’. In a famous lawsuit of the East Cree against developers, a lawyer representing the latter thought he had made a great point when one of the Indians was forced to admit he did not know how many rivers there were in the disputed territory. As Denny says, “The hunter knew every river in his territory individually and therefore had no need to know how many there were” (in Closs, p. 133). I am, as a matter of fact, not absolutely sure how many rooms there are in the (small) house where I am writing this chapter. Do you know how many chairs there are in the living room of the flat or house where you live?
As a rule we only bother to number and count things in quite special circumstances and when we know things or people very well we precisely abstain from counting because it is not only needless but may actually give offence — one does not, for example, count the number of paintings on the walls of a friend’s house. Of course, part of this movement from knowing things individually to numbering or counting them is a consequence of mass production. The standardized products of an industrial society cannot be known individually — they are depersonalised in much the same way as numbers are — so the only way to differentiate them is by giving them a label such as ’40-74-88’, or a bar code. This is basically the reason why people instinctively object to being given a number — rightly or wrongly they do not regard themselves as standardized social products.
‘Progress’ as it is generally conceived is a movement from the concrete to the abstract, from the qualitative to the quantitative, from the quantitative itself to the merely numerical — is this an unqualified ‘good thing’? I hardly think so even within the bounds of the most abstract science itself, mathematics. To the contemporary pure mathematician any relation to the physical everyday world is regarded as a sort of pollution. Even numbers themselves have been replaced by letters and the reduction of everything to algebra even includes the visual and tactile science of geometry : in a modern geometry textbook there are scarcely any figures and we prove theorems by messing about with co-ordinates. Lagrange actually boasted of writing a textbook on mechanics (of all subjects) which “contained not a single diagram”. There is something not only misguided but positively insulting in such an approach.
However, this is to jump the gun by several millennia, I am currently concerned to note the transition from the unnumerical hunting/foodgathering society to the highly numerate State bureaucracies that developed in the Middle East during the second millennium BC.5 Viewed from a distance individual differences between entities tend to disappear: trees which seen close up are clearly differentiable one from another, become so many dark shapes on the horizon and, as we move farther away still, are eventually reduced to a vague mass, the ‘many’ has become ‘one’. The hunter/pastoralist, and to a lesser extent the farmer, is so close to the important items of his life, is so immersed in the concrete, the here-and-now, that numerical reckoning is both largely unnecessary and conceptually difficult. The chief of a tribe would know every one of his male tribesmen personally whether by name or not, thus no need for counting. To the absolute despot who perhaps, like the later Chinese emperors never ventured outside the Forbidden City, his millions of subjects were not individualized but were simply a vague mass, the ‘people’. However, the State official who represented the Emperor was sufficiently distanced from the concrete to lose sight of individual distinctions but sufficiently aware of the need for accurate assessment of goods and services, to be in exactly the right position for a ‘numerical’ view of things. A centralized bureaucratic society at once gives rise to the need for numbering and arithmetic and makes the necessary awareness possible in the first place. As Denny puts it:

“The main condition under which arithmetical operations become useful is economic action at a distance. (…) The basic factors we have associated with the need for mathematics, increased alteration of the environment and increased dependence on others to perform specialized tasks, must have developed to the point where some people are specialized managers of the man-made agricultural system [and they are the ones] who direct the efforts of specialized workers.” (Closs, editor, p. 156)

It is worthwhile noting that commercial activity by itself, though it obviously does provide motivation for the development of arithmetic, is not in itself decisive. A good deal of trade until quite recently went on in terms of barter and even when ‘money’ was involved it was ‘concrete money’ not figures on a bank balance : medieval Japan used rice as a means of exchange and in Virginia during the seventeenth and early eighteenth centuries tobacco was employed. In Nigeriaquite extensive trade was conducted using bars of copper or iron and even bottles of gin as currency5.
When, however, we have a State currency and State controlled weights and measures imposed throughout a large area, we expect (and find) an accompanying development of sophisticated number systems and arithmetic operations as, for example, in Chinaunder the Han and later dynasties6. Arithmetic, along with writing itself, is the necessary and inevitable offspring of a centralized, bureaucratic society, the sort of society Wittfogel (who seems to have go ne out of fashion) characterized as ‘hydraulic’, i.e. based on State supply of water and other essential commodities.6 This is so because such societies necessarily involve ‘government from a distance’ and ‘quantitative assessment from a distance’, the precise opposite of conditions operating in tribal society.
Apart from wordless quantitative assessments ‘primitive’ man was perfectly able to indicate quantity other than by using number words or number marks. It would be ludicrous to suppose that a tribe possessing only the number words “One, two, many” was incapable of distinguishing between different larger quantities. Asked by the explorer Oldfield how many persons were killed in a certain tribal battle, the tribesman “answered by holding up his hand three times” (Conant, ). In such a case I doubt whether the tribesman was expressing our ‘15’ by finger counting : he was simply indicating a certain quantity ‘more or less’ equivalent to three times the fingers of one hand. In the majority of situations even today a rough assessment is adequate, indeed an exact numerical assessment often appears pedantic and thus objectionable. If asked how many people you were with in the pub last night you would perhaps say, “About half a dozen”. This is sufficient to distinguish the group from an intimate one consisting of two or three persons or an amorphous one consisting of twenty, and this is really all that one wants to know. One might call this ‘qualitative quantitative assessment’ and this is not a contradiction in terms. In the world of the hunter and foodgatherer distinction by type is so much more significant than distinction by number that even his or her quantitative assessments are largely qualitative. This is deeply shocking to the scientific mentality of today but the merit of such procedures is that they concentrate on what is important to the individual. They are in the current jargon ‘user-friendly’. I do not require to know the exact temperature it will be tomorrow — in fact I’d rather not know — but it might be useful to have an idea whether it is likely to feel cold, or conversely whether it is likely to be muggy, because I can them dress up appropriately. The expected exact temperature does not tell you what you want to know though the qualitative additional comments of the human weather reporter may do.

Notes

1 The only tolerably convincing case of ‘mathematics’ in the animal kingdom is the waggle dance of honey bees where a worker returning to the hive communicates information about the distance and direction of the food source by means of repetitive swaying movements. But see Animals as Mathematicians by Dr. Kalmus (Nature June 20, 1964 vol. 202). I would call most of the examples cited quasi-mathematical activity rather than ‘mathematical’ in the true sense but it depends how you define mathematics.

2 No slur on these ‘primitive’ peoples (or for that matter animals) is implied. Animals and hunting societies don’t use numbers much because they don’t need them and can’t be bothered with them. Well-intentioned contemporary writers who feel uneasy about their own implied superiority fall over backwards to try to persuade us that these ancient peoples did perhaps ‘use numbers’ after all and that animal species did evolve language ‘after all’. All the evidence is to the contrary: ‘primitive’ societies used altogether different quantitative methods which it would be misleading to call numerical.
Like Jean-Jacques Rousseau, I am often tempted to think that civilization (and possibly life itself) is a mistake : certainly I do not judge individuals or societies according to their technological and mathematical development. One could make out a strong case for the claim that writing and numbering skills are intimately associated with militarism and exploitation : after all it was the expansionist societies in the Middle Eastthat developed both. Napoleon was the metrifier of Europe, founded the prestigious Ecole Polytechnique and was no mean mathematician himself. Even today, America, though hardly a cultured country, is pre-eminent in mathematics. Amongst recent historians Innis is about the only one who sticks up for ‘oral’ societies as against tgose that employed writing. “[For Innis] writing, even before it was clearly mechanised, represented a mechanization of the spirit… Small was beautiful because it was built on a human scale of tongue and ear and living memory” (Godfrey, Foreword to Innis, Empire and Communication).

3 It has been shown experimentally that the maximum number of items a normal person can assess without counting them is about seven. Interestingly, the calculating prodigy Dase went up to twenty.

4 See Gay and Cole, The New Mathematics and an Old Culture, (New York, Holt, Rinehart & Winston 1967). Galton, a British nineteenth century explorer, remarks how the Damara people, a nomadic African tribe, were able to recall the faces of all the animals in a herd and thus detect not only how many animals were missing but the precise animal or animals (quoted in Zaslavsky, Africa Counts, p. 33).
Interestingly, the Kpelle were not so good at assessing numbers of people or huts in the village — probably because it was considered unlucky (or simply impolite?) to count people you actually knew.

5 “Bottles of gin passed from hand to hand for years without having been opened, and might represent the whole wealth of a chief. Basden reports seeing huge collections of gin bottles — a record of past transactions.” (Zaslavsky, Africa Counts Lawrence Hill 1973).

6 We are in fact today far closer to such societies than we are to the medieval feudal society where the lord lived on the land or to the commercial, slave owning societies of Ancient Greece. AlthoughAthens at its zenith controlled a wide-flung maritime empire, it never developed a powerful administrative class such as existed inPersia andEgypt. The mathematics of Ancient Greece was primarily geometry and the Greek number system compares unfavourably with that used by the earlier Babylonians who were magnificent calculators. Although geometry is necessary for surveying (and surveying necessary for taxation of land), centralised bureaucratic societies tend to be ‘arithmetical’ rather than ‘geometrical’. The vision of the administrator is a numerical vision, everything and everyone must have its number. Distinctions of type which include shape and colour are altogether secondary. The first society to impose wholesale decimalisation wasChina which, from the T’ang onwards, was controlled by an official class to which persons of any social origin (except merchants) had access, at least in theory. And Chinese mathematics, though very advanced, was not very geometrical either. It is notable — and alarming — that whereas Greek higher mathematics translated numbers into shapes by evolving a sort of geometrical algebra, the last two hundred years have seen shapes reduced to numbers in the arithmetic geometry of nineteenth century analysis and beyond.

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And I’d say that neither approach is innately “superior”, but tooled for different usages. Numerical approaches have value in some areas and non-numerical ones in others. For the simple tribes mentioned here, there may not have been anything which needed the numerical approach, and indeed it might be quite tedious and actually impractical for that purpose. Yet if we want to design an interplanetary spacecraft, clearly that approach is not going to cut the mustard by any stretch. So is either method “superior”? I don’t think so, I think that’s a worthless blanket judgment. Value judgments are often heavily context-dependent and should never be asserted as absolute. Each paradigm is superior in its correct context. Are hammers superior to screwdrivers? Of course, then you also have the notion that if all you have is a hammer, everything starts to look like a nail. 🙂

Is it “bad” to reduce shapes to number? Not if you can use it to design an airplane engine on a supercomputer. But is it “bad” to say this is the ONLY way and nothing else is useful EVER? Yes. But does anyone actually MAKE that claim? That’s what I’d be curious about. Is there a citation of such a claim?

With regards to diagrams in geometry — for Euclidean plane geometry, to me it seems that they should be maintained because they are very useful for getting the intuition. At the same time, however, the caveats must be mentioned and heeded so one does not draw the wrong conclusions from the diagram and one must be able to back up any insights from the diagram with the formal reasoning from axioms and theorems to conclusions (if this is possible at the level of knowledge available) since you are dealing with the idealized world specified by those axioms and also dealing with logic. Euclid, for example, made some assumptions which did not appear in the axioms, but rather were based on experiment with the diagrams. When you get up to the kind of geometry used in, say, General Relativity, however, they become much less useful.

And as for reducing to algebra — it’s good to know that you can always reduce to algebra, since that means it is always available to you as a tool, but it doesn’t mean it is necessarily the best way to go about solving a particular problem in actual practice. In theory, you can solve all Euclidean geometry algebraically, but in many cases this can be tedious. Applying the reduction to algebra mechanistically and without thought doesn’t solve problems.

As for this footnote: “I am often tempted to think that civilization (and possibly life itself) is a mistake” — by what condition? In an Atheist universe, life is just a natural process, it is no more a “mistake” than hydrogen fusing to helium in a star. In a Theistic belief system with a God-made universe, life is the creation of God and God is perfection and thus it is not a mistake. Mistake is the domain of humans. Humans made mistakes, neither nature nor God.

With regards to “certainly I do not judge individuals or societies according to their technological and mathematical development. ” Correct. They should instead be judged by how they utilize such things. Such things represent capacity and capability for both good and evil. Technological developments, for example, allow for our kids to stand a strong chance at surviving to adulthood, whereas before many would die painfully. This as much as they gave us guns by which kids could be painfully killed by deranged people (although on the whole, in technological societies the net effect is more kids survive now). It’s what we do with what we have that matters.

Also, though maybe it had been developed from militarism doesn’t mean it had to be, or that the alternative is get rid of warworld civilization and go back to dyingkidworld, or that it can only be used for militarism. If present civilization is too militaristic then we need a new form of civilization based on different principles, i.e. a path different from both stay-the-course-with-war-world-civilization and return-to-mass-child-death-world.